Peter C. Gibson

The Combinatorics of Scattering in Layered Media

Reflection and transmission of waves in piecewise constant layered media are important in various imaging modalities and have been studied extensively. Despite this, no exact time domain formulas for the Greens functions have been established. Indeed, there is an underlying combinatorial obstacle: the analysis of scattering sequences. In the present paper we exploit a representation of scattering sequences in terms of trees to solve completely the inherent combinatorial problem, and thereby derive new, explicit formulas for the reflection and transmission Greens functions.

Representation of Linear Operators by Gabor Multipliers

We consider a continuous version of Gabor multipliers: operators consisting of a short-time Fourier transform, followed by multiplication by a distribution on phase space (called the Gabor symbol), followed by an inverse short-time Fourier transform, allowing different localizing windows for the forward and inverse transforms. This chapter focuses on the following broad questions. Firstly, for a given pair of forward and inverse windows, which linear operators can be represented as a Gabor multiplier, and what is the relationship between the Kohn–Nirenberg symbol of such an operator and the corresponding Gabor symbol? We answer this question completely. Secondly, for a linear operator of a given type, can windows be specially chosen, or “tuned”, to suit the operator so that the Gabor symbol reflects the operator’s type? In studying this latter question for product-convolution operators, we derive a new class of “extreme value” windows that, with respect to the representation of linear operators, are more general than standard Gaussian windows while sharing many of Gaussian windows’ desirable properties. The results in this chapter help to justify techniques developed for seismic imaging that use Gabor multipliers to represent nonstationary filters and wavefield extrapolators.

Identification of minimum-phase-preserving operators on the half-line

Minimum phase functions are fundamental in a range of applications, including control theory, communication theory and signal processing. A basic mathematical challenge that arises in the context of geophysical imaging is to understand the structure of linear operators preserving the class of minimum phase functions. The heart of the matter is an inverse problem: to reconstruct an unknown minimum-phase-preserving operator from its value on a limited set of test functions. This entails, as a preliminary step, ascertaining sets of test functions that determine the operator, as well as the derivation of a corresponding reconstruction scheme. In this paper, we exploit a recent breakthrough in the theory of stable polynomials to solve the stated inverse problem completely. We prove that a minimum-phase-preserving operator on the half-line can be reconstructed from data consisting of its value on precisely two test functions. And we derive an explicit integral representation of the unknown operator in terms of these data. A remarkable corollary of the solution is that if a linear minimum-phase-preserving operator has rank at least 2, then it is necessarily injective.

Fourier Expansion of Disk Automorphisms via Scattering in Layered Media

A family of orthogonal polynomials on the disk (which we call scattering polynomials) serves to formulate a remarkable Fourier expansion of the composition of a sequence of Poincaré disk automorphisms. Scattering polynomials are tied to an exotic Riemannian structure on the disk that is hybrid between hyperbolic and Euclidean geometries, and the expansion therefore links this exotic structure to the usual hyperbolic one. The resulting identity is intimately connected with the scattering of plane waves in piecewise constant layered media. Indeed, a recently established combinatorial analysis of scattering sequences provides a key ingredient of the proof. At the same time, the polynomial obtained by truncation of the Fourier expansion elegantly encodes the structure of the nonlinear measurement operator associated with the finite time duration scattering experiment.

The structure of test functions that determine weighted composition operators

Operators that preserve minimum phase signals, or delayed minimum phase signals, have been shown to be important in practical signal processing contexts, and specifically in geophysical imaging, where one seeks to identify such operators using test signals. Which sets of test signals suffice to recover an unknown operator of the given type? In the present paper we answer this question by relating it to the identification of weighted composition operators acting on analytic functions on the disk. We provide an explicit parameterization of all minimal sets of test functions that identify weighted composition operators on the disk, and then apply the parameterization to construct realistic test signals for use in the geophysical context.

Acoustic imaging of layered media

Abstract This paper presents the echoes-to-impedance transform, a nonlinear transform designed for acoustic imaging of layered media—for example, sedimentary geological formations, biological tissue such as skin, or laminated structures in the built environment. The transform converts time domain digital reflection data directly into impedance as a function of spatial location, using minimal prior information about the source wavelet and no prepreprocessing. It is simple, fast, and, according to numerical experiments, highly accurate. More than this, physical structure is superresolved at a finer scale than that of the source wavelet. The derivation of the echoes-to-impedance transform stems from a recently developed numerical method for wave propagation in one dimension in conjunction with the theory of orthogonal polynomials on the unit circle.